In quantum field theory and statistical mechanics, the Mermin–Wagner theorem (also known as Mermin–Wagner–Hohenberg theorem, Mermin–Wagner–Berezinskii theorem, or Coleman theorem) states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2. Intuitively, this means that long-range fluctuations can be created with little energy cost and since they increase the entropy they are favored.
This is because if such a spontaneous symmetry breaking occurred, then the corresponding Goldstone bosons, being massless, would have an infrared divergent correlation function.
The absence of spontaneous symmetry breaking in d ≤ 2 dimensional systems was rigorously proved by David Mermin, Herbert Wagner (1966), and Pierre Hohenberg (1967) in statistical mechanics and by Sidney Coleman (1973) in quantum field theory. That the theorem does not apply to discrete symmetries can be seen in the two-dimensional Ising model.